
where \(h_{s_0}\)  is the bandwidth for the \(s_0^{\mbox{\tiny \rm th}}\)  model and the last equality can be deduced from the asymptotic distribution of semiparametric estimators under some regularity conditions [1]}. Because \(S_n^{1/2}n^{1/2}p_{\max }/(nh_{s_0}^4+h_{s_0}^{-1})\)  usually does not converge to 0 with general restrictions on \(h_{s_0}\) , Condition REF  is violated.
Note that this condition does not conflict with Condition REF  because they concern different distance measures. In particular, Condition REF  controls the distance between the estimators of \({\mu }_{(s)}\)  (i.e., \(\widehat{{\mu }}_{(s)}\)  and \(\widetilde{{\mu }}_{(s)}\) ) and their corresponding quasi-true values (i.e., \({\mu }_{(s)}^\ast \)  and \(\widetilde{{\mu }}_{(s)}^\ast \) ), while Condition REF  concerns the degree of misspecification which is the distance between the quasi-true value \({\mu }_{(s)}^\ast \)  and the true value \({\mu }\) .
